In those long ago days when Joe Cole emerged at West Ham, he was going to be the future of English football. It never quite worked out like that – it rarely does in England for players of his ilk – but on Sunday he returned to the East End and rediscovered the joy.

Cole began the afternoon in his now customary role as bench-warmer but injury gave him his chance and after a hesitant start he turned a match which had been shaping up to provide further confirmation of Liverpool's modern mid-table mediocrity. He also provided a hint that his glorious future may not all be behind him as his gamecraft enabled Liverpool to overcome the absence of a recognised striker.

Liverpool had arrived drained after an arduous journey back from playing Udinese on Thursday and without a striker, the suspension of Luis Suarez and Fabio Borini's foot injury robbing the manager, Brendan Rodgers, of his only two options. Rodgers' solution was to deploy Jonjo Shelvey in what he described as the "false nine" role popularised by Spain.

Shelvey came good in the end, forcing James Collins into own goal which enabled Liverpool to leapfrog West Ham into 10th, but for a long time he was a "false nine" only in the sense that he seemed an imposter in the centre-forward position. In the first half he played on the shoulder of West Ham's centre-halves like a conventional striker, but never looked comfortable. He closed defenders down, but he rarely held the ball up, nor did he drag his markers around. And when it came to threatening a goal, the one chance he did have, from a Stewart Downing cross in the final minute of the half, he stubbed wide. Meanwhile Shelvey's strengths, such as breaking forward from midfield, were lost to the team.

Then Cole, who was sent to the right-wing when he replaced Jose Enrique on 27 minutes (Downing going to left-back) began drifting inside, Liverpool became more fluid and West Ham's defence more porous. The Spanish "false nine" position could have been designed for Cole, but given difficulties on Merseyside it was no surprise Rodgers did not entrust him with the role. Indeed, when he came on he gave the ball away with his first three touches.

"He took time to get into game, but in the second half his quality came through," said Rodgers. Cole coming inside enabled Shelvey to drop deeper and affect the play ahead of him, and Downing to push on from left-back. West Ham were stretched and when they lost Mohamed Diamé to a hamstring tear they were broken.

With 15 minutes left Sterling combined with Shelvey, and Cole timed his run perfectly to score his first Premier League goal since April 2011. For a few seconds the joy and relief showed, then he remembered where he was – Hammers' fans gave him an ovation when he came on – and stilled his celebrations. Three minutes later, with West Ham reeling, Jordan Henderson crossed from the right and Collins deflected it over Jussi Jaaskelainen to hand Liverpool the points.

"We have to take it on the chin," said Sam Allardyce, who was as depressed with the prospect of losing Diamé for eight to 12 weeks as with losing the match. It does mean West Ham will not have to worry about losing Diamé in the January window but, said the manager, "I'd sooner have the aggravation of agents and people trying to buy him."

Rodgers's contrasting delight was palpable. "It's a brilliant win," he said. "Sam's made this a very difficult place to come. I thought for 25 minutes we were outstanding."

Liverpool had deservedly taken an 11th-minute lead. Glen Johnson, who had an excellent opening quarter, took a pass from Steven Gerrard, skipped inside Matt Jarvis, and rocketed a drive into the top left corner from 25 yards. Johnson, who began his career at West Ham, did not celebrate, something he should be practised at, having scored in his last three matches against them.

Gradually West Ham settled, and in one of the most significant developments Jarvis began to test Johnson's defending rather than vice-versa. The momentum of the game swung and Daniel Agger, having earlier rescued Enrique when he lost possession to Matt Taylor, had to make an excellent clearance as Carlton Cole shaped to convert a 27th-minute Jarvis cross. West Ham kept pressing with Shelvey and Gerrard booked for fouling Diamé, who had begun to usurp Gerrard's midfield dominance. Diamé also provided the breakthrough when, following excellent work by Mark Noble, the ball fell to him on the edge of the box and his volley struck Joe Allen's raised arm.

"The penalty knocked us back. We thought it was a wee bit unjust," said Rodgers. Seven minutes later a free-kick was played quickly out to the unmarked Jarvis and Gerrard inadvertently headed past his own goalkeeper. "A very good own goal," said Hammers' half-time announcer.

Although Liverpool controlled much of the second half it appeared West Ham would keep them at bay. When Cole's dummy enabled Sterling to run on to Downing's cross Jaaskelainen saved acrobatically, when Downing laid on a cross for Shelvey he headed wide. Then Diamé, whose half-time arrival turned last week's game against Chelsea, departed and Liverpool took full advantage.

Match facts

West Ham: JAASKELAINEN 6/10, DEMEL 5, REID 6, COLLINS 5, O’BRIEN 5, TAYLOR 6, DIAME 6, NOBLE 6, JARVIS 7, NOLAN 6, COLE 7

REINA 6, ENRIQUE 5, AGGER 5, SKRTEL 6, JOHNSON 7, ALLEN 6, GERRARD 6, LUCAS 6, DOWNING 6, SHELVEY 7 STERLING 8

Goals. West Ham United: Noble (pen) 36, Gerrard (og) 43. Liverpool: Johnson 11, Cole 76, Collins (og) 79

Substitutions: West Ham United McCartney 6 (Demel, 45), Tomkins (Diame, 73), Maiga (Taylor, 86). Liverpool Cole 7 (Enrique, 27), Henderson (Leiva, 71), Coates (Allen, 86). Bookings: West Ham United Taylor. Liverpool Shelvey, Gerrard. Man of the match Raheem Sterling. Match rating 8/10. Possession: West Ham United 39%. Liverpool 61%. Attempts on target: West Ham United 4. Liverpool 3. Referee Lee Probert (Wiltshire). Attendance 35,005